, Volume 8, Issue 2, pp 151-200

Shadows and traces in bicategories

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Abstract

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow”. In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.

Communicated by Ross Street.
K. Ponto and M. Shulman were supported by National Science Foundation postdoctoral fellowships during the writing of this paper.
© 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.